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How does an aspiring mathematician stay motivated in researching pure mathematics?

Hi! I am a high school senior set on becoming a "pure mathematician." Ever since I learned the proof to the quadratic formula in 10th grade, I have been chasing abstract mathematics and working to improve my problem-solving skills.
Less than a year ago, I came across a very interesting problem and have yet to solve it; many of the teachers I ask for help are stumped and leave the "discovery process" to me. I have worked on the problem for about 7 months but after making no significant progress, I decided to stop. Therefore, I ask: how do I motivate myself to continue? How do I know if my problem will lead to nowhere? How much time should I be devoting to this problem?

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Manuela’s Answer

Definitely read the book and watch the movie "The man who knew infinity". You will get a lot of inspiration.

IF you have the gene for math you will figure out in a blink of an eye if 3599 is a prime number or not by putting on an algebraic set of lens. You find the answer reading "Flash Boys" by Michael Lewis the author of "The Big Short", "MoneyBall" , "The Blind Side" and a few others. You saw the movie "The Big Short", didn't you?

You will be happy doing pure math but more fulfilled if you go on the applied side, meaning ... do something with the math.
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Yury’s Answer

This question puzzled me many times - how do you continue your research if you are not sure of the result? I guess, a short answer would be to study areas around the problem and see if somebody solved something similar or produced something that can be used as a building block.
I have a math doctorate degree and producing new results in math is not easy. Sometime you can achieve something by focusing on the task and sometime you really need to explore areas around. It would be better if your efforts are directed by professional mathematicians at least to some degree. And also, this sort of activity begs for math as a profession. With rare exceptions, great problem solvers have never solved just one problem but work on many problems after studying a lot of different areas. Many solutions are produced as by-products or extensions or failures to produce the desired result. A good example is NP-hard problems in CompSci - many people tried to create an algorithm for solving a problem and failed and concluded that there is a class of problems all of which are equivalent to each other and that was an achievement. Sounds like boring theoretical stuff but cryptography grew out of this and is now worth trillions.
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